# Tensor derivatives and gradients

**Next:** Index **Up:** Title page ** Previous:** General properties of tensors

The partial derivative of a tensor is not generally itself a tensor although it is for a scalar : is a 0/1 tensor.

It follows that transforms like a 1/1 tensor only if the second term is zero, i.e.

where and are constants. This is true for flat space with coordinates (*ct*,*x*,*y*,*z*) under the Poincaré transformations and in this case maybe interpreted as a 4D rotation matrix.

**Note however that it is not true for flat space with a general coordinate system or in the curved spacetime of General Relativity.**

We can define the derivative of a general *M*/*N* tensor along a curve parameterized by the proper time as follows:

If the basis vectors and one- forms are the same everywhere then:

where

and is the tangent to the curve. Thus is also like a *M*/*N* tensor, written as

where means . We can then define a *M*/*N*+1 tensor :

This is the tensor gradient [ remember the gradient of a scalar ].